Question

Solve the equation.cosec x = - √2

Answer 1

We know that principal value branch of

${cosec}^{ - 1} x \:\: is\:\: [-\frac{\pi}{2}, \frac{\pi}{2}]$-{0}

To solve the given equation we must keep consider that cosec and cosec inverse cancels each other only if x belongs to its principal value .
$cosec\: x = -\sqrt{2}\\ \\ x = {cosec}^{ - 1} ( -\sqrt{2} )\\ \\ we \: know \: that \: cosec \: \frac{\pi}{4} = \sqrt{2} \\ \\We\:\: know\:\:that\:\: \\ \\ {cosec}^{ - 1} ( - cosec\: x) = - {cosec}^{ - 1} ( cosec\: x) \\\\ So\\\\ x= {cosec}^{ - 1} (cosec ( -\frac{\pi}{4} )) \\ \\x=- {cosec}^{ - 1} (cosec ( \frac{\pi}{4} ))\\\\ x= - \frac{ \pi}{4} \: \: \: \: (x \: belongs \: to \: [- \frac{\pi}{2}, \frac{\pi}{2}])-(0)\\ \\ x = \frac{ - \pi}{4}$

is the solution of the equation.